Question

1. The size of claims made on an insurance policy are modelled through the following distribu- tion: You are interested in estimating the parameter λ > 0, using the following observations: 120, 20, 60, 70, 110, 150, 220, 160, 100, 100 (a) Verify that f is a density (b) Find the expectation of the generic random variable X, as a function of \ when A 1 (c) Prove that the method of moments estimator of λ is λι =斉. Calculate its observed value (d) Find the likelihood and loglikelihood functions (e) Find the score function (f) Find the observed Fisher information (g) Prove that the Maximum Likelihood Estinator (MLE) of λ is λ2 = 10% where y = Σ|-1 log (xi)/n. Calculate its observed value. (h) Find the expected Fisher information (i) Approximate the probability that the MLE takes value in the interval (λ-0.1, λ +0.1) G) Use a Gaussian density to approximate the likelihood function around the MLE, and use R to illustrate the approximation (include both the figure and the code used)

Answer 1

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